Question: All of the 3rd grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$2.50$ each for students, and the group paid $$33.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$24.00$ each for teachers and $$9.50$ each for students, and the group paid $$114.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+2.5y = 33.5}$ ${24x+9.5y = 114.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-24x-7.5y = -100.5}$ ${24x+9.5y = 114.5}$ Add the top and bottom equations together. $ 2y = 14 $ $ y = \dfrac{14}{2}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {8x+2.5y = 33.5}$ to find $x$ ${8x + 2.5}{(7)}{= 33.5}$ $8x+17.5 = 33.5$ $8x = 16$ $x = \dfrac{16}{8}$ ${x = 2}$ You can also plug ${y = 7}$ into $ {24x+9.5y = 114.5}$ and get the same answer for $x$ ${24x + 9.5}{(7)}{= 114.5}$ ${x = 2}$ There were $2$ teachers and $7$ students on the field trips.